91Ó°ÊÓ

\(\mathrm{T} / \mathrm{F}:\) The sum of continuous functions is also continuous.

Prove the given limit using an \(\varepsilon-\delta\) proof. $$ \lim _{x \rightarrow 2}\left(x^{3}-1\right)=7 $$

Approximate the given limits both numerically and graphically. $$ \lim _{x \rightarrow 3} \frac{x^{2}-2 x-3}{x^{2}-4 x+3} $$

Prove the given limit using an \(\varepsilon-\delta\) proof. $$ \lim _{x \rightarrow 2} 5=5 $$

Evaluate the given limits using the graph of the function. \(f(x)=x^{2} \sin (\pi x)\) (a) \(\lim _{x \rightarrow-\infty} f(x)\) (b) \(\lim _{x \rightarrow \infty} f(x)\)

Use the following information to evaluate the given limit, when possible. If it is not possible to determine the limit, state why not. $$ \begin{array}{ll} *\lim _{x \rightarrow 9} f(x)=6, & \lim _{x \rightarrow 6} f(x)=9, \quad f(9)=6 \\ *\lim _{x \rightarrow 9} g(x)=3, & \lim _{x \rightarrow 6} g(x)=3, \quad g(6)=9 \end{array} $$ $$ \lim _{x \rightarrow 6} f(g(x)) $$

Prove the given limit using an \(\varepsilon-\delta\) proof. $$ \lim _{x \rightarrow 0}\left(e^{2 x}-1\right)=0 $$

Approximate the given limits both numerically and graphically. $$ \lim _{x \rightarrow 2} \frac{x^{2}+7 x+10}{x^{2}-4 x+4} $$

Use the following information to evaluate the given limit, when possible. If it is not possible to determine the limit, state why not. $$ \begin{array}{ll} *\lim _{x \rightarrow 9} f(x)=6, & \lim _{x \rightarrow 6} f(x)=9, \quad f(9)=6 \\ *\lim _{x \rightarrow 9} g(x)=3, & \lim _{x \rightarrow 6} g(x)=3, \quad g(6)=9 \end{array} $$ $$ \lim _{x \rightarrow 6} g(f(f(x))) $$

Evaluate the given limits of the piecewise defined functions \(f\). \(f(x)=\left\\{\begin{array}{cl}x+1 & x \leq 1 \\ x^{2}-5 & x>1\end{array}\right.\) (a) \(\lim _{x \rightarrow 1^{-}} f(x)\) (c) \(\lim _{x \rightarrow 1} f(x)\) (b) \(\lim _{x \rightarrow 1^{+}} f(x)\) (d) \(f(1)\)